Recently developed transcatheter valves conform to a patient's anatomy while providing a uniform degree of rigidity and protection for critical valve components. They are capable of deforming circumferentially to adapt to the shape of the pre-existing valve annulus, but are not susceptible to deformation or migration due to normal movement of the heart. In one type of design, the valve prosthesis has a multi-level component that is anatomically shaped when deployed, thereby enhancing anchoring of the valve and reducing the risk of migration and paravalvular leaks. A particularly successful assembly of this type is disclosed in U.S. Pat. No. 7,914,569, the entire contents of which is incorporated by reference.